Parallel Tensor Arithmetic Applied to Multigrid Methods

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IP1

Algorithmic Adaptations to Extreme Scale

Algorithmic adaptations to use next-generation computers closer to their potential are underway. Instead of squeez- ing out ﬂops the traditional goal of algorithmic optimality, which once served as a reasonable proxy for all associated costs algorithms must now squeeze synchronizations, memory, and data transfers, while extra ﬂops on locally cached data represent only small costs in time and energy.

After decades of programming model stability with bulk synchronous processing, new programming models and new algorithmic capabilities (to make forays into, e.g., data as- similation, inverse problems, and uncertainty quantification) must be co-designed with the hardware. We briefly recap the architectural constraints, then concentrate on two kernels that each occupy a large portion of all scientific computing cycles: large dense symmetric/Hermitian systems (covariances, Hamiltonians, Hessians, Schur com- plements) and large sparse Poisson/Helmholtz systems (solids, fluids, electromagnetism, radiation diffusion, grav- itation). We examine progress in porting solvers for these kernels to the hybrid distributed-shared programming en- vironment, including the GPU and the MIC architectures that make up the cores of the top scientific systems on the floor and on the books. How will the hierarchical solvers that lead in scalability (e.g., fast multipole, hierarchically low rank matrices, multigrid) map onto the more rigidly programmed and less reliably performant structures within a node?

David E. Keyes KAUST

david.keyes@kaust.edu.sa IP2

Challenges for Climate and Weather Prediction in the Era of Exascale Computer Architectures: Os- cillatory Stiﬀness, Time-Parallelism, and the Role of Long-Time Dynamics

For weather or climate models to achieve exascale performance on next-generation heterogeneous computer architectures they will be required to exploit on the order of hundred-million-way parallelism. This degree of parallelism far exceeds anything possible in todays models even though they are highly optimized. In this talk I will discuss one of the mathematical issues that leads to the limitations in space- and time-parallelism for climate and weather prediction models oscillatory stiﬀness in the PDE of the form:

∂u

∂t +1

L(u) +N(u,u) =D(u), u(0) =u0,

where the linear operator L has pure imaginary eigenvalues, the quadratic nonlinear term isN(u,u)Drepresents dissipation. The is a small non-dimensional parameter.

The operator⁻¹Lresults in time oscillations on an order O() time scale, and generally requires small time steps for explicit methods, and implicit methods if accuracy is required. I will discuss the case whenis ﬁnite and two algorithms: 1) a fast-converging HMM-parareal-type method and 2) a time-parallel matrix exponential.

Beth Wingate University of Exeter B.Wingate@exeter.ac.uk IP3

Ingredients for Computationally Eﬃcient Solution

of Large-Scale Image Reconstruction Problems Image reconstruction problems provide great opportuni- ties to bring together many scientiﬁc computing techniques to advance the state of the art in inverse problems and in disciplinary areas. First, we need to tailor the inverse problem to the application to produce the regularized solution while considering how a priori knowledge is enforced.

We might choose to enforce hard constraints, like non- negativity, sparsity and/or high-contrast. Alternatively, we may choose to employ a learned dictionary or a parameterized image model that imposes those constraints directly on image space and simultaneously restrict the search space. Second, those modeling choices, which are in- teresting problems in and of themselves, necessitate the use of sophisticated optimization algorithms. Third, since each optimization step requires multiple forward model simulations, techniques from the multigrid, model reduction and randomization communities need to be explored to attain the maximum computational eﬃciency. In this talk, we provide an overview of some of these scientiﬁc computing techniques that have been successfully used in image reconstruction, and provide some combinations of techniques that have led to particularly fruitful outcomes in the context of a few applications.

Misha E. Kilmer

Mathematics Department Tufts University

misha.kilmer@tufts.edu

IP4

Making Sense of our Universe with Supercomput- ers

In computational cosmology and astrophysics we encounter some of the most complex multi-scale and multi-physics problems. In the past decades, algorithmic advances have enabled ever more realistic numerical models of a very wide range of astrophysical objects. These range from stars to galaxies, from planets to the large scale structure of the Universe, from molecular clouds to star clusters, from su- pernovae explosions to super-massive black holes in centers of galaxies. We routinely create three dimensional models of how our Universe may have originated, how its structure formed, how the very ﬁrst stars and galaxies came about, how pulsars work, and how black holes merge and generate gravitational waves to just name a few such applications.

We will highlight some examples of three particular algorithmic breakthroughs and the particular advances and in- sights they have enabled so far. These describe adaptive mesh reﬁnement simulations capturing 15 orders of magnitude in length scale, adaptive ray tracing for high accuracy radiation hydro-dynamical simulations, as well as a new noise-free approach to solve the collision-less Boltz- mann equation of interest in cosmology as well as in plasma physics. We will also present the scientiﬁc visualizations created from these simulations. These have been shown on various television programs, international planetarium shows and numerous print media.

Tom Abel

Stanford University Hi@TomAbel.com

IP5

Multiscale Modelling: Powerful Tool or Too Many

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Promises

Multiscale modelling aims to provide systematic linking of different time and length scales in order to enable simulations at different levels of spatial and/or temporal resolu- tions. No single unique method exists or is even foresee- able and hence the choice of the most appropriate method or mapping depends on the properties of interest. The roots of multiscale simulations go back to the 1960s and 1970s, but the last decade has been brought them in the mainstream of method development and as a viable approach to model complex systems. It has been applied to viral capsids, fluid flow, crystal growth, proteins, col- loids, and polymers to mention a few examples. Current techniques range from pragmatic, such as using solubili- ties for force field mapping to algorithmic, using equilib- rium particle data for solving an inverse problem (using methods such as Inverse Boltzmann, force matching and Inverse Monte Carlo), particle-continuum coupling and using Langevin and Fokker-Planck equations and mapping procedure. Procedures such as GENERIC also extend the multiscale approach to irreversible processes. In this talk, I will discuss multiscale methods from the perspective of soft materials based on our own work, provide perspectives for future development and problems involving multiscaling.

Mikko Karttunen

Eindhoven University, The Netherlands mkarttu@gmail.com

IP6

Adaptive and Multiscale Methods in Subsurface Flow Modeling

In reservoir simulation, the ratio of the largest scale to the smallest scale is typically very large. The smaller scales can not easily be upscaled or neglected as they can have (significant) impact on reservoir ﬂow. At the same time, decision making under uncertainty requires fast and accurate ﬂow simulations for often large ensembles of model realizations.

No wonder then that the reservoir simulation community has developed many adaptive as well as multiscale methods to reduce runtime. Especially the last decade has seen a number of exciting new approaches, which I will discuss here, as well as the outstanding challenges that remain.

Margot Gerritsen

Dept of Energy Resources Engineering Stanford University

margot.gerritsen@stanford.edu IP7

Productive and Sustainable: More Eﬀective Com- putational Science and Engineering

Computational Science and Engineering (CSE) is effective to the extent it contributes to overall scientific and engineering pursuits. Its contributions are most tangible when delivering concrete scientific and engineering results via modeling, simulation and analysis. At the same time, de- livery of CSE results is impacted by how we develop and support the ecosystem that produced these results, including, in particular, software and people. While delivering results is the ultimate goal of our CSE efforts, the short- est path to results is often not the most productive and sustainable. In this presentation we discuss elements that impact the effectiveness of CSE efforts, beyond just the direct activities to produce results. We discuss how processes, tools and a holistic view of efforts can lead to more

eﬀective CSE. We also discuss the importance of human factors in CSE activities, highlighting ways we can provide natural incentives toward more eﬀective CSE.

Michael Heroux

Sandia National Laboratories maherou@sandia.gov

IP8

Computational Science and Engineering Achieve- ments in the Designing of Aircraft

This presentation will give an overview of what Computa- tional Science and Engineering has brought in design ca- pacities these last years in the aeronautics industry. The unceasing increase in computing resources and remarkable improvements of solving methods have enabled industry to perform computations that were not conceivable several before. An emphasis will be scoped to optimization methods as actual engineering tools utilized for industrial applications, in particular for aerodynamic shape design.

Numerical formulation and implementation issues will be recalled and illustrations of applications will be discussed.

The study of eﬃcient multidisciplinary approaches will be also addressed. New ﬁeld of applications of Computational Science and Engineering have emerged these last years.

Stochastic methods are in the process to take more and more an important place in the toolset of the design en- gineer and beyond. Some examples of application will be given. The presentation will end with the challenges re- lated to Computational Science and Engineering for aero- nautical industry.

Bruno Stouﬄet

Dassault Aviation, France

Bruno.Stouﬄet@dassault-aviation.com IP9

Stochastic Gradient Methods for Machine Learning The stochastic gradient method has emerged as the most powerful technique for training the large-scale statistical models that form the foundation of modern machine learning systems. This talk provides an accessible introduction to the mathematical properties of stochastic gradient methods, and the intuition behind them. To set the stage, we present two case studies, one involving sparse linear models for text classiﬁcation and one involving deep neural networks for image recognition. We then discuss advanced algorithms that control noise and make use of second order information. We conclude the talk with a discussion of the geometry of deep neural networks.

Jorge Nocedal

Department of Electrical and Computer Engineering Northwestern University

j-nocedal@northwestern.edu CP1

Theories and Algorithms of Integrated Singular Value Decomposition (iSVD)

The singular value decomposition (SVD) is an important tool in many applications. However, the computational cost of traditional algorithms for solving SVD grows rapidly as the data size increasing. Randomized SVD provides a method to randomly sketch a matrix and ﬁnd its approximate low-rank SVD with lesser resources. Some

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techniques can be used to improve the accuracy of a single sketching. Instead of focusing on one single sketching, iSVD is an algorithm to improve the accuracy by integrating multiple simple random sketching. The main idea of the proposed algorithm is solving a restricted optimization problem for a suitable objective function. The Kolmogorov-Nagumo-type average is used to solve this optimization problem. Some numerical results, including the application on 1000 Genomes data, will also be presented.

Ting-Li Chen

Institute of Statistical Science Academia Sinica

tlchen@stat.sinica.edu.tw Dawei D. Chang

Institute of Applied Mathematical Sciences National Taiwan University

davidzan830@gmail.com Su-Yun Huang

syhuang@stat.sinica.edu.tw Hung Chen

Institute of Applied Mathematical Sciences National Taiwan University

hchen@math.ntu.edu.tw Chienyao Lin

youyuoims94@gmail.com Weichung Wang

National Taiwan University

Institute of Applied Mathematical Sciences wwang@ntu.edu.tw

CP1

An Eﬃcient Iterative Penalization Method Based on Recycled Krylov Subspaces and Its Application to Impulsively Started Flows

The Vortex Particle-Mesh (VPM) method is well suited for solving advection dominated incompressible ﬂows. How- ever, the eﬃcient and accurate handling of solid bound- aries in this method is still an active domain of research.

The Brinkman penalization method embeds the object in the ﬂuid domain and enforces the velocity inside the ob- stacle to be u = u_b, where u_b is the desired velocity.

This additional constraint is added to the vorticity form of the incompressible Navier-Stokes equations through a La- grange relaxation. The boundary enforcement conditions the capture of vorticity production at the wall and is thus paramount to the accuracy of the global method. Hejle- sen et al. proposed to ﬁrst solve the unpenalized Navier- Stokes equations and then, to impose the constraint using a Jacobi-like iterative process. In this work, we formulate the penalization problem inside a VPM method as a linear system to solve at every time step. Recovering the velocity from the vorticity (i.e. solving a Poisson problem) makes the matrix-vector product highly expensive. We use a recycling iterative solver, rBiCGStab, to solve it in order to reduce the number of iterations. This method is validated against a benchmark ﬂow past a cylinder (

mathrmRe= 550) and then, we assess the computational gain with a ﬂow past a cylinder and a plate (Re = 9500

and

mathrmRe= 1000, respectively).

Thomas Gillis

Universite catholique de Louvain

Institute of Mechanics, Materials and Civil Engineering thomas.gillis@uclouvain.be

CP1

Preconditioning Irk Methods for Time-Dependent Fluid Flow Problems

We examine block preconditioners for time-dependent incompressible Navier-Stokes problems. In some time- dependent problems, explicit time stepping methods can require much smaller time steps for stability than are needed for reasonable accuracy. This leads to taking many more time steps than would otherwise be needed. With implicit time stepping methods, we can take larger steps, but at the price of needing to solve large linear systems at each time step. We consider implicit Runge-Kutta (IRK) methods. Suppose our PDE has been linearized and discretized with N degrees of freedom. Using an s-stage IRK method leads to ansN×sN linear system that must be solve at each time step. These linear systems are block s×ssystems, where each block isN×N. We investigate preconditioners for such systems, taking advantage of the structure of the subblocks.

Victoria Howle Texas Tech

victoria.howle@ttu.edu CP1

Polynomial Preconditioned Arnoldi for Eigenvalues Polynomial preconditioning has been explored for Krylov methods but has not become standard. In this talk, we look at the Arnoldi method for eigenvalues and give a simple choice for the polynomial preconditioner. It is shown that this approach can significantly improve the efficiency for difficult problems.

Jennifer A. Loe Baylor University jennifer loe@baylor.edu Ronald Morgan

Department of Mathematics Baylor University

ronald morgan@baylor.edu CP1

Linear Equations and Eigenvalues Using Krylov Methods on Multiple Grid Levels

We wish to solve large systems of linear equations and large eigenvalue problems. We try to combine the efficiency of using coarse grids with the power of Krylov subspaces. For linear equations, this involves a two-grid approach that de- flates eigenvalues on the fine grid using approximate eigenvectors computed on the coarse grid. For eigenvalue problems, eigenvectors from coarse grids can be improved on finer grids using a variant of restarted Arnoldi. These methods are more robust than standard multigrid and sometimes are much more efficient than standard Krylov methods.

Ronald Morgan

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Department of Mathematics Baylor University

ronald morgan@baylor.edu CP1

A Factored ADI Method for Sylvester Equations with High Rank Right Hand Sides

The factored alternating direction implicit (ADI) method is a technique used to solve Sylvester equations of the form AX−XB = M N^T, where M and N are tall-and- skinny matrices. In this talk, we develop a variation of the ADI method that improves performance when M N^T is of medium to high rank, specifically for the application of solving elliptic partial differential equations. In particular, we employ it to solve elliptic partial differential equations on the disk expressed in a low rank format.

Heather D. Wilber Boise State University hdw27@cornell.edu Alex Townsend Cornell University townsend@cornell.edu CP1

Parallel Implementations of Integrated Singular Value Decomposition (iSVD)

Integrated Singular Value Decomposition (iSVD) is an algorithm for computing low-rank approximate singular value decomposition of large size matrices. The iSVD inte- grates diﬀerent low-rank SVDs obtained by multiple random subspace sketches and achieve higher accuracy and better stability. While iSVD takes higher computational costs due to multiple random sketches and the integration process, these operations can be parallelized to save computational time. We parallelize the algorithm for mul- ticore/manycore hybrid CPU-GPU clusters. We propose algorithms and data structures to increase the scalability and reduce communication. With parallelization, iSVD can solve matrices with huge size, and achieve near-linear scalability with respect to the matrix size and the number of machines. We implement the algorithms in C++, with several techniques used so that many tuning decisions can be determined at compile time to reduce run-time over- head. Some huge size examples will be presented to show the performance of the implementation.

Mu Yang

Institute of Applied Mathematical Sciences, National Taiwan

muyang@ntu.edu.tw

Su-Yun Huang, Ting-Li Chen Institute of Statistical Science Academia Sinica

syhuang@stat.sinica.edu.tw, tlchen@stat.sinica.edu.tw Weichung Wang

National Taiwan University

Institute of Applied Mathematical Sciences wwang@ntu.edu.tw

CP1

A Fast Direct Solver for Elliptic PDEs on Locally-

Perturbed Domains

Many problems in science and engineering can be formulated as integral equations with elliptic kernels. In particular, in optimal control and design problems, the domain geometry evolves and results in a sequence of discretized linear systems to be constructed and inverted. While the systems can be constructed and inverted independently, the computational cost is relatively high. In the case where the change in the domain geometry for each new problem is only local, i.e. the geometry remains the same except within a small subdomain, we are able to reduce the cost of inverting the new system by reusing the pre-computed fast direct solvers of the original system. The resulting solver only requires inexpensive matrix-vector multiplica- tions, thus dramatically reducing the cost of inverting the new linear system. Numerical results will illustrate the performance of the solver.

Yabin Zhang

Dept. of Computational and Applied Mathematics Rice University, Houston, TX

YZ89@RICE.EDU Adrianna Gillman Rice University

Department of Computational and applied mathematics adrianna.gillman@rice.edu

CP2

Hierarchical Model Reduction for Incompressible Flows in Pipes

The Hierarchical Model Reduction (HiMod) is a novel technique for the efficient solution of fluid problems in pipes that joins computational efficiency with numerical accuracy. According to this method, the transverse dynamics is represented in terms of a generalized Fourier modal ex- pansion, whose axial dependence is discretized via a Finite Element Method. In such a way the original problem results as a system of coupled 1D problems. The power of this technique lies in its hierarchical nature, as the accuracy can be tuned by a proper selection of the number of transverse modes. Aiming at real medical applications, we apply HiMod to scalar (Advection-Diffusion Reaction models) and vector problems (incompressible Navier-Stokes equations) on 3D cylindrical domains. The application of the technique to such geometries is non-trivial, especially in the respect of the identification of a basis function set.

We numerically assess diﬀerent options. Patient-speciﬁc blood-vessels geometries are handled by appropriate geo- metrical maps. Numerical tests point out the capability of HiMod to detect the transverse dynamics of the physical phenomenon, as opposed to other approaches that rely on averaging the transverse dynamics in a purely 1D setting.

Our method tries to find a practical trade-off between the accuracy of 3D modeling and the efficiency of 1D modeling.

Soﬁa Guzzetti, Alessandro Veneziani

Department of Mathematics and Computer Science Emory University

soﬁa.guzzetti@emory.edu, avenez2@emory.edu Simona Perotto

MOX - Modeling and Scientiﬁc Computing Dipartimento di Matematica

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simona.perotto@polimi.it CP2

Inertial Conﬁnement Fusion Simulations Using a Front Tracking API

The Stony Brook front tracking code FronTier has been ex- tracted into an Application-Programming Interface (API) for easy implementation into external physics codes. Front tracking is a well validated algorithm which shows improved accuracy relative to experiment for fluid mixing applications. In this talk we detail the first use of this API through implementation in the University of Chicago code FLASH. We detail the process required for implementation and discuss the benefits of the coupling of a front tracking algorithm for fluid mixing problems such as Rayleigh-Taylor and Richtmyer-Meshkov instabilities. Our main application is the use of the front tracking API for the simulation of Inertial Confinement Fusion (ICF) cap- sules. We present 2D simulations in a spherical geometry and discuss the impact of front tracking on ICF simulations. We show that for coarser grids, the front tracking simulations are closer to a converged result, a key require- ment for the heavy computational requirements associated with ICF simulations.

Jeremy A. Melvin

University of Texas - Austin jmelvin@ices.utexas.edu Verinder Rana

Stony Brook University Stony Brook NY vrana@ams.sunysb.edu Ryan Kaufman SUNY at Stony Brook rkaufman@ams.sunysb.edu James Glimm

Stony Brook University glimm@ams.sunysb.edu CP2

Conservative DEC Discretization of Incompressible Navier-Stokes Equations on Arbitrary Surface Sim- plicial Meshes With Applications

A conservative discretization of incompressible Navier- Stokes equations over surfaces is developed using discrete exterior calculus (DEC). The mimetic character of many of the DEC operators provides exact conservation of both mass and vorticity, in addition to superior kinetic energy conservation. The employment of various discrete Hodge star deﬁnitions based on both circumcentric and barycen- tric dual meshes is also demonstrated. Some of the investigated deﬁnitions allow the discretization to admit arbitrary surface simplicial meshes instead of being limited only to Delaunay meshes, as in previous DEC-based discretizations. The discretization scheme is presented in detail along with several numerical test cases demonstrat- ing its numerical convergence and conservation properties.

The developed scheme is also applied to explore the curva- ture eﬀects on ﬂow past a circular cylinder.

Mamdouh S. Mohamed

Physical Sciences & Engineering Division, KAUST, Jeddah, KSA

mamdouh.mohamed@kaust.edu.sa

Anil Hirani

Department of Mathematics

University of Illinois at Urbana-Champaign, IL, USA hirani@illinois.edu

Ravi Samtaney KAUST

ravi.samtaney@kaust.edu.sa CP2

Stability of Oscillatory Rotating Boundary Layers Some of the most popular applications of fluid mechanics come in aerodynamics, and methods of laminar flow control on swept wings have become increasingly important over the past few decades; especially with the global emphasis on emissions reduction. Another reason for the recent development of this field is the availability of high- performance computers, meaning calculations that would have been impossible only a decade ago can now be performed quickly on a workstation. With instability mecha- nisms in common with a swept wing, the rotating disk has long been considered as a valid approximation to this flow and is also far more amenable to experiments. The experimental setup for a rotating disk study requires a much smaller space and much less expensive equipment than the wind tunnel required for a swept wing experiment. For this reason, there are a wealth of experimental and theoretical studies of the rotating disk boundary layer and this talk will extend these established results. A recent study by Thomas et. al. [Proc. R. Soc. A (2011) 467, 2643-2662]

discusses adding an oscillatory Stokes layer to a channel flow and shows some stabilising results. We present a similar modification to the rotating disk configuration by way of periodic modulation and provide results from both direct numerical simulations and local eigenvalue analyses showing a stabilising effect.

Scott N. Morgan Cardiﬀ University MorganSN@cardiﬀ.ac.uk CP2

Marker Re-Distancing and Sharp Reconstruction for High-Order Multi-Material Front Dynamics A new method for high-order front evolution on arbitrary meshes is introduced. The method is a hybrid of a La- grangian marker tracking with a Discontinuous Galerkin projection based level set re-distancing. This Marker-Re- Distancing (MRD) method is designed to work accurately and robustly on unstructured, generally highly distorted meshes, necessitated by applications within ALE-based hydro-codes. Since no PDE is solved for re-distancing, the method does not have stability time step restrictions, which is particularly useful in combination with AMR, used here to efficiently resolve fine interface features. A high-order (implemented up to the 6th-order) level set approach is utilized for a new sharp treatment of mix elements, which reconstructs multiple-per-element solution fields (one for each material present in the mix element).

Reconstruction incorporates interfacial jump conditions, which are enforced in the least-squares sense at the interfacial marker positions provided by MRD. Since no explicit diﬀerentiation across the interface is involved in the as- sembly of residuals for mass, momentum and energy equations, the method is capable of capturing discontinuous solutions at multi-material interfaces with high order, and without Gibbs oscillations. The method performance is

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demonstrated on a number of numerical tests, including well-known benchmarks, and phase-change ﬂuid dynamics problems relevant to the selective laser melting applications.

Robert Nourgaliev, Patrick Greene, Sam Schoﬁeld LLNL

nourgaliev1@llnl.gov, greene30@llnl.gov, schoﬁeld5@llnl.gov

CP2

Direct Computations of Marangoni-Induced Flows Using a Volume of Fluid Method

The volume of ﬂuid (VOF) interface tracking methods have been used for simulating a wide range of interfacial ﬂows.

An improved accuracy of the surface tension force computation has enabled the VOF method to become widely used for simulating surface tension driven ﬂows. We present a new method for including variable surface tension in a VOF based Navier-Stokes solver. The tangential gradient of the surface tension is implemented using an extension of the classical continuum surface force model that has been pre- viously used for constant surface tension simulations. Our method can be applied for computing the surface gradients of surface tension that is temperature or concentration dependent.

Ivana Seric, Shahriar Afkhami New Jersey Institute of Technology is28@njit.edu, shahriar@njit.edu Lou Kondic

Department of Mathematical Sciences, NJIT University Heights, Newark, NJ 07102 kondic@njit.edu

CP2

Physics-Based Preconditioning for a High-Order Rdg-Based Compressible Flow Solver with Phase Change

The numerical simulation of flows associated with metal additive manufacturing processes such as selective laser melting and other laser-induced phase change applications present new challenges. Specifically, these flows require a fully compressible formulation with phase change, since rapid density variations occur due to laser-induced melting and solidification of metal powder.

We investigate the preconditioning for a recently developed all-speed compressible Navier-Stokes solver that addresses such challenges. The equations are discretized with a re- constructed Discontinuous Galerkin method and integrated in time with fully implicit discretization schemes. The resulting set of non-linear and linear equations are solved with a robust Newton-Krylov (NK) framework.

To enable convergence of the highly ill-conditioned linearized systems, we employ a physics-based operator split preconditioner (PBP), which utilizes a robust Schur com- plement matrix for the velocity-pressure and velocity- temperature block-systems. We investigate different options of splitting the physics (field) blocks as well as different iterative solvers to approximate the action of the inverse of the preconditioned system. We demonstrate that our PBP-NK framework is scalable and converges for high CFL/Fourier numbers on classic problems in fluid dynamics as well as for laser-induced phase change problems.

Brian Weston

University of California, Davis

Lawrence Livermore National Laboratory brianweston@gmail.com

Robert Nourgaliev LLNL

nourgaliev1@llnl.gov Jean-Pierre Delplanque University of California, Davis delplanque@ucdavis.edu CP3

Multigrid Preconditioned Lattice Boltzmann Method Based on Central Moments for Eﬃcient Computation of Fluid Flows

Lattice Boltzmann (LB) Method is one of the more recent promising developments in computational fluid dynamics (CFD) for based on a local kinetic approach flow simulations. Like other classical explicit time-marching methods, the LBM can suffer from slow convergence rate to steady state, especially at low Mach numbers. This is due to the relatively large disparity between the acoustic wave and fluid convection speeds, i.e. high eigenvalue stiffness, which can be alleviated by preconditioning the LB method. Fur- thermore, we combine such a preconditioned LB scheme based on the efficient stream-and-collide procedure with the multigrid method based on the nonlinear full approximation storage scheme to provide convergence acceleration at an optimal cost. The collision step is formulated using central moments to provide enhanced numerical stability.

In addition, we develop consistent inter-grid transfer operators based on using extended moment equilibria in the collision step involving a tunable parameter that keeps the flow properties such as viscosities, and hence flow physics, invariant across different grid levels. Finally, we validate our new multigrid preconditioned LB method for various benchmark problems and then demonstrate the significant improvements achieved in convergence acceleration, by factors of one or more orders of magnitude, for various sets of the preconditioning parameters and Reynolds numbers and Mach numbers.

Farzaneh Hajabdollahi, Kannan Premnath University of Colorado Denver

farzaneh.hajabdollahiouderji@ucdenver.edu, kannan.premnath@ucdenver.edu

CP3

Multigrid Preconditioning for Space-Time Dis- tributed Optimal Control of Parabolic Equations This work is concerned with designing optimal order multigrid preconditioners for space-time distributed control of parabolic equations. The focus is on the reduced problem resulted from eliminating state and adjoint variables from the KKT system. Earlier numerical experiments have shown that our ability to design optimal order preconditioners depends strongly on the discretization of the parabolic equation, with several natural discretizations leading to suboptimal preconditioners. Using a continuous- in-space-discontinuous-in-time Galerkin discretization we obtain the desired optimality.

Mona Hajghassem, Andrei Draganescu Department of Mathematics and Statistics University of Maryland, Baltimore County

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mona4@umbc.edu, draga@umbc.edu CP3

Spectral Matrix Analysis of the Semi-Discrete Compressible Navier-Stokes Equations Using Large-Scale Eigensolvers

Implicit integration methods for the compressible Navier- Stokes equations rely on iterative methods for sparse linear systems. Code performance depends upon the physics, models, boundary conditions, and numerical methods. At large scale and high order of accuracy, performance is largely determined by the linear solver which in turn depends upon the base iterative method, preconditioner selection, and eigenstructure of the linear problems constructed through spatial discretization. To greater un- derstand the impact of algorithm choices on linear solver and preconditioner performance and design, we use large- scale eigensolvers to obtain a partial spectrum (collection of eigenvalues) of the discretization matrix. We perform this analysis on a high-order entropy-stable spectral collocation method for the compressible Navier-Stokes equations on laminar and turbulent problems.

Michael A. Hansen University of Utah

Sandia National Laboratories mike.hansen.utah@gmail.com Travis Fisher

Sandia National Labs tcﬁshe@sandia.gov CP3

Block Triangular Preconditioners for Linearization Schemes of the Rayleigh-B´enard Convection Prob- lem

In this work, we compare two block triangular preconditioners for different linearizations of the Rayleigh-Bénard convection problem discretized with finite element methods. The two preconditioners differ in the nested or non- nested use of a certain approximation of the Schur comple- ment associated to the Navier-Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time.

This investigation reveals some inconsistencies in the lit- erature that are hereby discussed. We observe that the non-nested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard-Newton scheme. Numerical results of two- and three-dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.

Guoyi Ke

Texas Tech University guoyi.ke@ttu.edu Eugenio Aulisa

Department of Mathematics and Statistics.

Texas Tech University eugenio.aulisa@ttu.edu Giorgio Bornia

Dept. of Mathematics and Statistics Texas Tech University

giorgio.bornia@ttu.edu Victoria Howle Texas Tech

victoria.howle@ttu.edu CP3

Comparison of Techniques for Hermitian Inte- rior Eigenvalue Problems: Reﬁned, Harmonic and Polynomial Filters

Polynomial and rational polynomial ﬁltered methods have gained popularity for large interior eigenproblems.

FEAST-like solvers exhibit good performance when fac- torizing the operators is affordable. For larger problems, recent research in Chebyshev polynomials and inexact FEAST have proposed heuristics to tune performance sensitive parameters such as the selection of the contour points, the polynomial degree, or the tolerance for the approximate solution of the linear systems. However they are still hard to apply in black-box solvers. We present a modification of the refined extraction technique of Zhongxiao Jia, that is at least as robust but far less expensive, especially when using block methods. The new extraction in combination with Jacobi-Davidson has shown good results in the context of solving difficult interior eigenvalue problems arising in the computation of singular triplets.

An important advantage of Jacobi-Davidson is that can be seen as a rational polynomial filtered method, which offers a natural way to adjust dynamically the shifts and the tolerance to solve linear systems. In this talk we provide experimental results with Hermitian problems com- paring: Jacobi-Davidson with the new refined extraction as included in the last version of PRIMME, 2.0; polynomial filtered Lanczos in the recent released EVSL; and inexact solution of linear solvers with the FEAST software.

Eloy Romero Alcalde

Computer Science Department College of Williams & Mary eloy@cs.wm.edu

Andreas Stathopoulos College of William & Mary Department of Computer Science andreas@cs.wm.edu

CP3

Fast Algorithms for Jacobi Matrices from Modiﬁ- cation by Rational Functions

Given a Jacobi matrix for a sequence of orthogonal polynomials with respect to some measuredλ(x), the goal of this project is to generate the Jacobi matrix for modification of that measuredλ(x) obtained from˜ dλ(x) by multiplying by a rational function. Through partial fraction decomposition, this amounts to modification by dividing by linear or irreducible quadratic factors. The proposed method re- verses the algorithm for modifying by multiplying factors, due to Golub and Kautsky, combined with a root finding iteration such as the secant method. The entire process requires only O(n) floating point operations compared to O(n³) floating point operations for the inverse Cholesky algorithm due to Elhay and Kautsky. One application is to obtain Jacobi matrices for generalized Jacobi polynomials.

Amber C. Sumner

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The University of Southern Mississippi amber.sumner@usm.edu

James V. Lambers

University of Southern Mississippi Department of Mathematics James.Lambers@usm.edu

CP3

Estimating Matrix Bilinear and Quadratic Forms Using Krylov Subspace Methods with Recycling Matrix bilinear forms C^TA⁻¹B, for a nonsymmetric ma- trix A, and quadratic formsB^TA⁻¹B or its trace, for symmetric positive deﬁnite A, frequently appear in applications. We show how to evaluate or estimate these bilinear and quadratic forms accurately and cheaply using Krylov subspace methods with recycling. We demonstrate the effectiveness of our approach on three applications: functional error estimation and mesh adaptation in CFD, tomography, and topology optimization. This is joint work with Chris Roy and Will Tyson (VT) on CFD applications, Misha Kilmer (Tufts) on Tomography, and Xiaojia Zhang and Glaucio Paulino (Georgia Tech) on Topology Optimization.

Katarzyna Swirydowicz, Eric De Sturler Virginia Tech

kswirydo@vt.edu, sturler@vt.edu William Tyson, Christopher Roy VT

wtyson45@vt.edu, cjroy@vt.edu Misha E. Kilmer

Mathematics Department Tufts University

misha.kilmer@tufts.edu Xiaojia Zhang, Glaucio Paulino Georgia Tech

xzhang645@gatech.edu, paulino@gatech.edu

CP3

Why Are So Many Matrices in Computational Sci- ence of Low Rank?

In computational mathematics, matrices and functions that appear in practice are so often of surprisingly low rank and this structure is often expertly exploited. Since random (“average”) matrices are almost surely of full rank, mathematics needs to explain the remarkable abundance of low-rank structures in computational mathematics. In this talk, we will give a new characterization of low-rank matrices, which we use to explore why (1) Droplets on a pond, (2) Non-equally sampling of functions, (3) Elliptic partial diﬀerential equations and (4) The world ﬂags, all lead to low-rank objects.

Alex Townsend Cornell University townsend@cornell.edu Gil Strang

MIT

gilstrang@gmail.com CP4

Computing the Ankle-Brachial Index with Compu- tational Fluid Dynamics

Peripheral artery disease (PAD), in which narrowing and blockage of peripheral arteries reduces blood flow to the extremities of the body, is associated with a six-fold increase in mortality risk from cardiovascular disease. PAD is diagnosed by computing the ankle-brachial index (ABI), a metric relating blood pressure in the ankles and up- per arms. With parallel computing, we use 3D computational fluid dynamics to simulate flow in a complete, patient-derived arterial system and compute the ABI. The simulations employ a massively parallel CFD application, HARVEY, designed for large-scale hemodynamic simulations and based on the lattice Boltzmann method. Sim- ulations were conducted on Vulcan, a Blue Gene/Q su- percomputer at Lawrence Livermore National Laboratory with 393,216 cores. We consider the dependence of ABI on simulation resolution and find adequate numerical convergence at 50μm. The influence of body posture on ABI is investigated by incorporating gravitational forces corre- sponding to supine and standing body positions. Addi- tionally, we consider the influence of an aortic coarctation, which imposes the same hemodynamic compromise on the peripheral arteries as PAD, and observe the expected de- crease in ABI.

John Gounley

Old Dominion University john.gounley@duke.edu Erik W. Draeger

Lawrence Livermore Nat. lab.

draeger1@llnl.gov Jane Leopold

Brigham and Women’s Hospital leopold@partners.org¿

Amanda Randles Duke University

amanda.randles@duke.edu CP4

Dynamic Mesh Adaptation for Front Evolution Us- ing Discontinuous Galerkin Based Weighted Con- dition Number Relaxation

A new mesh smoothing method designed to cluster cells near a dynamically evolving interface is presented. The method is based on weighted condition number mesh relaxation with the weight function being computed from a level set representation of the interface. The weight function is expressed as a Taylor series based discontinuous Galerkin (DG) projection, which makes the computation of the derivatives of the weight function needed during the condition number optimization process a trivial mat- ter. For cases when a level set is not available, a fast method for generating a low-order level set from discrete cell-centered ﬁelds, such as a volume fraction or index function, is provided. Results show that the low-order level set works equally well for the weight function as the actual level set. The method retains the excellent smoothing capabilities of condition number relaxation, while providing a method for clustering mesh cells near regions of inter-

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est. Dynamic cases for moving interfaces will show that the new method is capable of maintaining a desired resolution near the interface with an acceptable number of relaxation iterations per time step, which demonstrates the method’s potential to be used as a mesh relaxer for arbitrary Lagrangian Eulerian (ALE) methods. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Information management release number LLNL-ABS-702781.

Patrick Greene, Samuel Schoﬁeld, Robert Nourgaliev LLNL

greene30@llnl.gov, schoﬁeld5@llnl.gov, nour- galiev1@llnl.gov

CP4

A DPG Method for Viscoelastic Fluids

We propose discontinuous Petrov-Galerkin (DPG) ﬁnite element method for the steady-state Oldroyd-B equations.

The method is attractive because of its built-in stability–

that is, no stabilization terms need to be added to the system–as well as its built-in error estimator which can be used for adaptive mesh generation. Notable, as well, is that the method will always result in a symmetric, positive defi- nite stiffness matrix. We exploit each of these properties in our analysis and perform verification upon a confined cylinder benchmark problem. The code was written in C++ and available online with the Camellia finite element software [1]. [1] Roberts, N. V. (2014). Camellia: A software framework for discontinuous PetrovGalerkin methods. Comput.

Math. Appl., 68(11):15811604.

Brendan Keith

The University of Texas at Austin

The Institute for Computational Engineering and Sciences brendan@ices.utexas.edu

Philipp Knechtges

Chair for Computational Analysis of Technical Systems RWTH Aachen University

knechtges@cats.rwth-aachen.de Nathan Roberts

Argonne National Laboratory nvroberts@alcf.anl.gov Stefanie Elgeti

RWTH Aachen University elgeti@cats.rwth-aachen.de Marek Behr

RWTH Aachen University

Chair for Computational Analysis of Technical Systems behr@cats.rwth-aachen.de

Leszek Demkowicz

Institute for Computational Engineering and Sciences (ICES)

The University of Texas leszek@ices.utexas.edu CP4

Large Time Step HLL and HLLC Schemes

We present the Large Time Step (LTS) extension of the Harten-Lax-van Leer (HLL) and Harten-Lax-van Leer Con-

tact (HLLC) schemes. The LTS methods are a class of explicit methods whose time step is not restricted by the classical Courant-Friedrichs-Lewy (CFL) condition. The basic idea is to increase the domain of dependence by modifying the numerical ﬂux function, and allowing linear interaction of waves from diﬀerent Riemann problems. Such schemes were developed by Randall LeVeque [R.J. LeVeque. Large time step shock-capturing techniques for scalar conservation laws, SIAM J. Numer. Anal., 19:1091-1109, 1982] in the nineteen eighties. The original scheme and the suc- cessive versions have been developed and applied mostly within a framework of the Godunov scheme and Roe’s approximate Riemann solver. We show that it is possible to construct the LTS extension of the HLL and HLLC schemes. We apply the LTS HLL and HLLC schemes to a number of test cases for inviscid gas dynamics, such as shock tube, double rarefaction and Woodward-Colella blast wave problem. It is shown that the schemes yield results comparable to those of standard and LTS Roe scheme. In addition, we show that both LTS HLL and HLLC schemes preserve positivity for a double rarefaction test case where standard Roe and LTS Roe schemes fail and that both schemes yield entropy satisfying resolution of the rarefaction waves.

Marin Prebeg, Bernhard M¨uller

Norwegian University of Science and Technology marin.prebeg@ntnu.no, bernhard.muller(at)ntnu.no CP5

Multigrid Kss Methods for Time-Dependent PDEs Krylov Subspace Spectral (KSS) methods are traditionally used to solve time-dependent, variable-coeﬃcient PDEs.

Lambers, Cibotarica, and Palchak improved the eﬃciency of KSS methods by optimizing the computation of high- frequency components. This talk will demonstrate how one can make KSS methods even more eﬃcient by using a multigrid-like approach for low-frequency components.

The essential ingredients of multigrid, such as restriction, residual correction, and prolongation, are adapted to the time-dependent case. Numerical experiments demonstrate the eﬀectiveness of this approach.

Haley Dozier

The University of Southern Mississippi Haley.Dozier@usm.edu

James V. Lambers

University of Southern Mississippi Department of Mathematics James.Lambers@usm.edu CP5

Automatic Construction of Scalable Time-Stepping Methods for Stiﬀ Pdes

Krylov Subspace Spectral (KSS) Methods have been demonstrated to be highly scalable time-stepping methods for stiff nonlinear PDEs. However, ensuring this scalability requires analytic computation of frequency-dependent quadrature nodes from the coefficients of the spatial differential operator. This talk describes how this process can be automated for various classes of differential operators to facilitate public-domain software implementation.

Vivian A. Montiforte

The University of Southern Mississippi vivian.mclain@usm.edu

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James V. Lambers

University of Southern Mississippi Department of Mathematics James.Lambers@usm.edu CP5

Multigrid Preconditioning of Linear Systems Aris- ing in the Semismooth Newton Solution of Dis- tributed Optimal Control of Elliptic Equations with State Constraints

The purpose of this research is to design eﬃcient multigrid preconditioners for distributed optimal control of elliptic equations. In this talk we focus on preconditioning of linear systems arising in the semismooth Newton solution of distributed control for elliptic equations with state- constraints. This research is building upon our earlier work on the associated control-constrained problems. Analytical and numerical results are presented.

Jyoti Saraswat

Department of Mathematics & Physics Thomas More College

saraswj@thomasmore.edu Andrei Draganescu

Department of Mathematics and Statistics University of Maryland, Baltimore County draga@umbc.edu

CP5

Inexact Algebraic Factorization Methods for the Steady Incompressible Navier-Stokes Equations The steady Navier-Stokes problem is of fundamental importance in science and engineering. Although there has been much research on the topic through the decades, the problem remains a challenging one due to its nonlinearity and the large size of its associated linear systems. Because of its importance and diﬃculty, new techniques for reducing the problems computational burden remain in demand.

For the time-dependent problem, there exists a class of popular methods for solving the Navier-Stokes problem efficiently known as Inexact Algebraic Factorization Meth- ods (see e.g. [ A. Quarteroni, F. Saleri, and A. Veneziani, Factorization methods for the numerical approximation of Navier-Stokes equations, Comp. Meth. in App. Mech., 188 (2000), pp. 505-526]). These methods work by approx- imating the saddle-point system with an inexact block LU factorization. These methods exhibit good accuracy and stability properties while significantly reducing the costs associated with solving, assembling, and storing the linear system. In this work, we extend these methods to the steady problem by showing that the stiffness matrix can be used as a suitable approximation to the nonlinear term under certain conditions. Numerical results in 2D and 3D are then discussed and presented.

Alex Viguerie Emory University aviguer@emory.edu Alessandro Veneziani

MathCS, Emory University, Atlanta, GA ale@mathcs.emory.edu

CP5

Investigations of Several Mhd Solvers Based on

Discontinuous Galerkin Finite Element Method Compressible ﬂow with magnetic phenomena occurs in a wide variety of scientiﬁc and engineering applications.

The eﬃcient parallel numerical simulation for these problems is very important. In this paper, several 1D and 2D Hydro-Dynamics (HD) and Magneto-Hydro-Dynamic (MHD) solvers have been developed based on Discontin- uous Galerkin method on Finite Element method. Their performance has been investigated and compared. These include various Riemann solvers for capturing the shocks and contact waves. Furthermore new method has been used to capture the shocks, and it has been compared to above traditional solvers. These solvers are in parallel version, benchmark and simulations will be shown, and some conclusions will be drawn from them.

Xiaohe Zhufu

Institute of Software, Chinese Academy of Science xiaohe@iscac.ac.cn

Yanfei Jiang

Center for Astrophysics, Harvard University yanfei.jiang@cfa.harvard.edu

Zhaoming Gan, Defu Bu, Maochun Wu

Shanghai Astronomical Observatory, Chinese Academy of Scienc

zmgan@shao.ac.cn, dfbu@shao.ac.cn, maochun@ustc.edu.cn

Jin Xu

Institute of Software, ISCACS Chinese Academy of Science, China xu jin@iscas.ac.cn

CP6

DOF-Reducing Small-Lebesgue Polygonal Spectral Basis Functions with Application to Discontinuous FEM

A closed form relation is proposed to approximate Fekete points (a Nondeterministic Polynomial (NP) problem) on a general convex/concave polyhedral. The approximate points are used to generate basis functions using the SVD of the Vandermonde matrix. Arbitrary order nodal, orthogonal and orthonormal polygonal basis functions are derived. It is shown that these basis are the best choice to enforce minimum DOF while maintaining a small Lebesgue constant when very high p-reﬁnement is done. The proposed basis are rigorously proven to achieve arbitrary order of accuracy by satisfying Weierstrass approximation theo- rem in R^d. The practicality of these basis are evaluated in Discontinuous Galerkin (DG) and Discontinuous Least- Squares (DLS) formulations. The accuracy, stability and DOF reduction is demonstrated for the linearized acoustics and two-dimensional compressible Euler equations on some benchmark problems including a cylinder, airfoil, vortex convection and compressible vortex shedding from a triangle.

Arash Ghasemi

SimCenter: National Center for Computational Engineering

ghasemi.arash@gmail.com CP6

A Hybrid Adaptive Compressible/Low-Mach-

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Number Method

Flows in which the primary features of interest do not rely on high-frequency acoustic effects, but in which long- wavelength acoustics play a nontrivial role, present a computational challenge. Integrating the entire domain with low Mach number methods would remove all acoustic wave propagation, while integrating the entire domain with the fully compressible equations would be prohibitively expensive due to the CFL time step constraint. For example, thermoacoustic instabilities might require fine resolution of the fluid/chemistry interaction but not require fine resolution of acoustic effects, yet one does not want to neglect the long-wavelength wave propagation and its interaction with the larger domain. The proposed lecture will present a new hybrid algorithm that has been developed to address these type of phenomena. In this new approach, the fully compressible Navier-Stokes equations are solved on the entire domain, while their low Mach number counterparts are solved on a subregion of the domain with higher spatial resolution. The coarser acoustic grid communicates inho- mogeneous divergence constraints to the finer low Mach number grid, so that the low Mach number method allows the long-wavelength acoustics. We will demonstrate the effectiveness of the new method on practical cases such as the aeroacoustics generated by the vortex formation in an unstable low-Mach mixing layer.

Emmanuel Motheau

Lawrence Berkeley National Lab

Center for Computational Sciences and Engineering emotheau@lbl.gov

Ann S. Almgren

Lawrence Berkeley National Laboratory asalmgren@lbl.gov

John B. Bell CCSE

Lawrence Berkeley Laboratory jbbell@lbl.gov

CP6

An Arbitrary High Order Imex Scheme For Ex- tended Magnetohydrodynamics Equations Using Entropy Conservative Flux

XMHD is an extended plasma fluid model which assumes quasineutrality and differs from the existing reduced models by the formulation of Generalized Ohm’s Law. Previous models for example Ideal MHD or Hall MHD are not suf- ficient to describe all the dynamics encountered in general plasma due to neglected terms in their derivation. In particular to capture low density current, inclusion of electron inertia terms is necessary. The Ohm’s Law for XMHD has been modified from resistive MHD Ohm’s Law through the introduction of electron pressure, electron inertia and Hall term allowing ion and electron demagnetization. In this work we proposed a finite difference scheme that uses an entropy conservative flux with an appropriate numerical diffusion operator for the simulation of the fluid part and rest of the system is treated with a Local Lax Friedrich flux. For the divergence constraints of Maxwell equation to be explicitly satisfied, correction potential forBandE has been enforced. The IMEX idea where source term is treated implicitly and flux is treated explicitly increases the efficiency of the scheme by reducing numerical cost since using the special structure of the source term we will show that we only need to solve a system of 9 linear equations at each grid point explicitly. The scheme performs well when

two ﬂuid eﬀects are important.

Chhanda Sen

Indian Institute of Technology Delhi chhanda27sen@gmail.com

Harish Kumar

Indian Institute of Technology Delhi, India hkumar[at]maths.iitd.ac.in

CP6

Energy Conservation Moment Method to Solve the Multi-Dimensional Vlasov-Maxwell-Fokker-Planck Equations

We present a numerical method to solve the Vlasov- Maxwell-Fokker-Planck (VMFP) system using the regularized moment method proposed in [Z. Cai, R. Li, Nu- merical regularized moment method of arbitrary order for Boltzmann-BGK equation, SIAM J. Sci. Comput (2010), Z. Cai, Y. Fan, R. Li, Globally hyperbolic regularization of Grad’s moment system in one dimensional space, Comm.

Math. Sci (2013)]. In [Y. Wang, S. Zhang, CICP (ac- cepted)], the globally moment system to the 1D Vlasov- Poisson-Fokker-Planck (VPFP) is deduced and the numerical scheme to keep the balance law of the total momentum is provided. However, it cannot keep the conservation of total energy. In this paper, the moment model of the electromagnetic field term is derived and then, we extend the global moment system to the multi-dimensional VMFP and VPFP systems, where the electromagnetic field term and the Fokker-Planck collision term are reduced into the linear combination of the moment coefficients. The Strang- splitting method is adopted to solve the whole moment system, which is splitted into the conservation part and the MFP part. Most importantly, a special semi-implicit numerical scheme which could keep the conservation of total energy is proposed to solve the Maxwell’s equations at the MFP part. The time evolution of the solutions to both 2D VMFP and VPFP systems are studied to demonstrate the stability and accuracy of the regularized moment method when applied to the VMFP system.

Yanli Wang

Institute of Applied Physics and Computational Mathematics

wang yanli@iapcm.ac.cn CP7

Adjoint-Enabled Optimization and UQ for Radia- tion Shield Design

Radiation shields make commodity microelectronics practical for use in satellite and other space systems. Shield designers wish to take advantage of new materials and manufacturing processes to meet strict weight limits while protecting electronics from naturally occurring pro- ton and electron radiation environments. Our work couples Sandia National Laboratories Dakota software (http://dakota.sandia.gov) with its SCEPTRE radiation transport code to automate the design exploration and reliability analysis process, enabling analysts to evaluate prospective shield materials and geometries. This talk highlights eﬃciency gains from pairing gradient-based optimization and uncertainty quantiﬁcation algorithms in Dakota with newly implemented adjoint sensitivities in SCEPTRE.

Brian M. Adams

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Sandia National Laboratories

Optimization/Uncertainty Quantiﬁcation briadam@sandia.gov

CP7

Ensemble Kalman Filtering for Inverse Optimal Control

Solving the inverse optimal control problem for discrete- time nonlinear systems requires the construction of a sta- bilizing feedback control law based on a control Lyapunov function (CLF). However, there are few systematic approaches available for defining appropriate CLFs. We propose an approach that employs Bayesian filtering methodology to parameterize a quadratic CLF. In particular, we use the ensemble Kalman filter to estimate parameters used in defining the CLF within the control loop of the inverse optimal control. Results are demonstrated on a real-world application to mathematical biology.

Andrea Arnold

North Carolina State University anarnold@ncsu.edu

Hien Tran

Center for Research in Scientiﬁc Computation North Carolina State University

tran@ncsu.edu CP7

Interpolatory Model Reduction of Parameterized Bilinear Dynamical Systems

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been already successfully extended to nonparametric bilinear dynamical systems. However, this is not the case for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce the conditions that projection subspaces need to satisfy in order to obtain parametric tangential interpolation of each subsystem transfer functions. These conditions also guarantee that the parameter gradient of each subsystem transfer function is matched tangentially by the parameter gradient of the correspond- ing reduced order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including extra vectors in the projection subspaces. As in the linear case, for two-sided projections, the basis construction does not require computing neither the gradient nor the Hessian to be matched.

Andrea Carracedo Rodriguez

Virginia Polytechnic Institute and State University (Virginia Tech)

crandrea@vt.edu Serkan Gugercin Virginia Tech

Department of Mathematics gugercin@vt.edu

CP7

Domain Decomposition Algorithms for Uncer- tainty Quantiﬁcation: High-Dimensional Stochas-

tic Systems

Domain decomposition (DD) algorithms developed by Sub- ber for uncertainty quantification of large-scale stochastic PDEs are extended using multi-level preconditioned conju- gate gradient methods (PCGM) to enhance its capability to tackle high-dimensional stochastic systems. These DD algorithms will be further tuned using Anderson acceleration method to expedite the convergence. Parallel sparse matrix-vector operations are used to cut floating-point operations and memory requirements. Both numerical and parallel scalabilities of these algorithms are presented for a diffusion equation having spatially varying diffusion coeffi- cient modeled by a non-Gaussian stochastic process.

Abhijit Sarkar Associate Professor

carleton University, Ottawa, Canada abhijit sarkar@carleton.ca

Ajit Desai

Carleton University, Canada ajit.desai@carleton.ca Mohammad Khalil

Sandia National Laboratories mkhalil@sandia.gov

Chris Pettit

United States Naval Academy, USA pettitcl@usna.edu

Dominique Poirel

Royal Military College of Canada, Canada poirel-d@rmc.ca

CP7

An Asymptotic-Preserving Stochastic Galerkin Method for the Semiconductor Boltzmann Equa- tion with Random Inputs and Diffusive Scalings In this talk, I will introduce the generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method for the linear semi-conductor Boltzmann equation with random inputs and diffusive scalings. The random inputs are due to uncertainties in the collision kernel or initial data. We study the regularity of the solution in the random space, and prove the spectral accuracy of the gPC-SG method. We then use the asymptotic- preserving framework for the deterministic counterpart developed in [Jin] to come up with the stochastic asymptotic-preserving gPC-SG method for the problem under study, which is efficient in the diffusive regime. Numerical experiments will be presented to validate the accuracy and asymptotic properties of the method. This is a joint work with Prof Shi Jin. [Jin]: Discretization of the multi scale semiconductor Boltzmann equation by diffusive relaxation schemes, S. Jin and L.Pareschi, J. Comput. Phys., 161: 312-330, 2000.

Liu Liu

University of Wisconsin Madison lliu84@wisc.edu

CP7

Bayesian Model Reduction for Nonlinear Dynamics Using Automatic Relevance Determination

The concept of automatic relevance determination (ARD) is invoked in this study to select models of nonlinear dy-

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namical systems in the form of stochastic ordinary diﬀeren- tial equations (ODEs). The Bayesian method can provide misleading results when the prior probability distributions are assigned arbitrarily to a subset (some) of parameters for which no information is available. In such cases, ARD provides an automatic model selection scheme to identify the optimal model nested under an overly complex model.

Given noisy measurement data, a relatively complicated model is envisioned to represent the dynamical system.

Then a model selection problem is posed to ﬁnd the best model nested under the envisioned model. This problem is transferred from the model parameter space to a hyper- parameter space by imposing a parametrized prior distribution called the ARD prior. The parameters of the prior distribution are known as hyper-parameters; in practice, they explicitly capture the relevance of model parameters.

This approach regularizes the adaptation of the posterior distribution to the data so as to avoid overﬁtting.

Abhijit Sarkar Associate Professor

carleton University, Ottawa, Canada abhijit sarkar@carleton.ca

Rimple Sandhu

Carleton University, Canada rimple sandhu@carleton.ca Chris Pettit

United States Naval Academy, USA pettitcl@usna.edu

Mohammad Khalil

Sandia National Laboratories mkhalil@sandia.gov

Dominique Poirel

Royal Military College of Canada, Canada poirel-d@rmc.ca

CP7

A Computational Bayesian Framework to Paral- lelize an Adaptive Markov Chain Monte Carlo This paper introduces a method to optimize the convergence of a new adaptive Markov Chain Monte Carlo approach (AMCMC) needed to formulate high dimensional parametric Bayesian formulations. The proposed approach relies on the sampling of parallel chains to ensure the capturing of all modes of the posterior distribution by developing a two-step synchronous sampling mechanism.

The full integration of the posterior distribution apply- ing the Bayesian paradigm using MCMC and Metropolis- Hasting (MH) algorithms is known to be computational inefficient. AMCMC presents a deterministic tuning of the proposal distribution in a two-step process, to optimize the acceptance ratio and expedite the MCMC convergence via several independent runs. A methodology to randomly generate parallel combinations of MCMC is proposed, which aim is to search for the optimized chains acceptance ratio. A comparison between the proposed and conventional MCMC-MH algorithms is discussed, when these are applied to a forward model simulating shale gas well-production, presented with real borehole production of daily observations, required to complete a probabilistic model calibration. The proposed approach is validated by its use on different well production data, showing significant computational efficiency, but most importantly, the promise for the method to be fully Bayesian parallelized

(work in progress).

Yasser Soltanpour Texas A&M University yasser soltanpour@tamu.edu Zenon Medina-Cetina

Stochastic Geomechanics Laboratory Texas A&M University, TX, USA zenon@tamu.edu

CP7

Stochastic Dirichlet Boundary Optimal Control of Steady Navier-Stokes Equations

When a physical system under control includes a stochastic component, the construction, modeling, and analysis of the controls become much more difficult.? As a specific case, we consider the optimal control of a system governed by the Navier-Stokes equations with a stochastic Dirich- let boundary condition.? Control conditions applied only on the boundary are associated with reduced regularity, as compared to internal controls. To ensure existence of solution and efficiency of numerical simulations, the stochastic boundary conditions are required to belong almost surely toH¹(∂D),? similar to theH-valued? infinite dimensional Wiener process. To simulate the system, state solutions will be approximated using the stochastic collocation finite element approach, and sparse grid techniques are applied to the boundary random field. The one shot optimality systems are derived from the Lagrange functional. Error estimates are computed for the optimality almost surely using samples, and for the state equation using interpo- lated boundary conditions.? Error estimates for the adjoint equations are derived from a duality argument, and the control equation comes via a non-conforming finite element variational crime. A numerical simulation can then be made, using a combination of Monte Carlo and sparse grid methods, which demonstrates the efficiency of the algorithm.

Wenju Zhao

Department of Scientiﬁc Computing Florida State University

wz13@my.fsu.edu Max Gunzburger Florida State University

Department of Scientiﬁc Computing mgunzburger@fsu.edu

CP8

Compact Implicit Integration Factor Method for Solving High Order Diﬀerential Equations

Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential equations. Implicit integration method (IIF) method along with its compact form (cIIF), which treats spatial derivatives exactly and reaction terms implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reaction and spatial derivatives. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. The compact representation for IIF (cIIF) was introduced to save the computational cost and storage for this purpose. Another chal-

Parallel Tensor Arithmetic Applied to Multigrid Methods

Sponsored by the SIAM Activity Group on Computational Science and Engineering (CSE)

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